4.18.25 $-ay(x)y^{\prime }(x)+b+x{y}^{\prime }(x{)}^2=0$

ODE
$$-ay(x)y^{\prime }(x)+b+x{y}^{\prime }(x{)}^2=0$$ ODE Classification

[[_homogeneous, `class G`], _rational, _dAlembert]

Book solution method
No Missing Variables ODE, Solve for $y$

Mathematica ✓
cpu = 0.635193 (sec), leaf count = 243

$$\{\text{Solve}[\frac{-2a{\tanh }^{-1}\left(\frac{\sqrt{a^{2}y(x{)}^2-4bx}}{ay(x)}\right)-2(a-1){\tanh }^{-1}\left(\frac{\sqrt{a^{2}y(x{)}^2-4bx}}{y(x)-ay(x)}\right)+a\log ((1-2a)y(x{)}^2+4bx)-\log ((1-2a)y(x{)}^2+4bx)+a\log (4bx)}{2a-1}=c_1,y(x)],\text{Solve}[\frac{2a{\tanh }^{-1}\left(\frac{\sqrt{a^{2}y(x{)}^2-4bx}}{ay(x)}\right)+2(a-1){\tanh }^{-1}\left(\frac{\sqrt{a^{2}y(x{)}^2-4bx}}{y(x)-ay(x)}\right)+a\log ((1-2a)y(x{)}^2+4bx)-\log ((1-2a)y(x{)}^2+4bx)+a\log (4bx)}{2a-1}=c_1,y(x)]\}$$

Maple ✓
cpu = 0.023 (sec), leaf count = 106

$$\{[x\left(\mathit{\_}\mathit{T}\right)={\mathit{\_}\mathit{T}}^{\frac{1}{a}{(1-a^{-1})}^{-1}}(\frac{b}{{\mathit{\_}\mathit{T}}^{{(a-1)}^{-1}}(2a-1){\mathit{\_}\mathit{T}}^2}+\mathit{\_}\mathit{C}\mathit{1}),y\left(\mathit{\_}\mathit{T}\right)=\frac{1}{\mathit{\_}\mathit{T}a{\mathit{\_}\mathit{T}}^{{(a-1)}^{-1}}(2a-1)}(2(\mathit{\_}\mathit{C}\mathit{1}{\mathit{\_}\mathit{T}}^{{(a-1)}^{-1}}{\mathit{\_}\mathit{T}}^2+b)(a-1/2){\mathit{\_}\mathit{T}}^{{(a-1)}^{-1}}+{\mathit{\_}\mathit{T}}^{{(a-1)}^{-1}}b)]\}$$ Mathematica raw input

DSolve[b - a*y[x]*y'[x] + x*y'[x]^2 == 0,y[x],x]

Mathematica raw output

{Solve[(-2*a*ArcTanh[Sqrt[-4*b*x + a^2*y[x]^2]/(a*y[x])] - 2*(-1 + a)*ArcTanh[Sq
rt[-4*b*x + a^2*y[x]^2]/(y[x] - a*y[x])] + a*Log[4*b*x] - Log[4*b*x + (1 - 2*a)*
y[x]^2] + a*Log[4*b*x + (1 - 2*a)*y[x]^2])/(-1 + 2*a) == C[1], y[x]], Solve[(2*a
*ArcTanh[Sqrt[-4*b*x + a^2*y[x]^2]/(a*y[x])] + 2*(-1 + a)*ArcTanh[Sqrt[-4*b*x + 
a^2*y[x]^2]/(y[x] - a*y[x])] + a*Log[4*b*x] - Log[4*b*x + (1 - 2*a)*y[x]^2] + a*
Log[4*b*x + (1 - 2*a)*y[x]^2])/(-1 + 2*a) == C[1], y[x]]}

Maple raw input

dsolve(x*diff(y(x),x)^2-a*y(x)*diff(y(x),x)+b = 0, y(x),'implicit')

Maple raw output

[x(_T) = _T^(1/a/(1-1/a))*(b/(_T^(1/(a-1)))/(2*a-1)/_T^2+_C1), y(_T) = (2*(_C1*_
T^(1/(a-1))*_T^2+b)*(a-1/2)*_T^(1/(a-1))+_T^(1/(a-1))*b)/_T/a/(_T^(1/(a-1)))/(2*
a-1)]